Word Problem Solving of Students with Autistic Spectrum Disorders and Students with Typical Development
Young Seh Bae
Submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy under the Executive Committee of the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY 2013
© 2013 Young Seh Bae All rights reserved
ABSTRACT
Mathematical Word Problem Solving of Students with Autistic Spectrum Disorders and Students with Typical Development
Young Seh Bae
This study investigated mathematical word problem solving and the factors associated with the solution paths adopted by two groups of participants (N=40), students with autism spectrum disorders (ASDs) and typically developing students in fourth and fifth grade, who were comparable on age and IQ ( >80). The factors examined in the study were: word problem solving accuracy; word reading/decoding; sentence comprehension; math vocabulary; arithmetic
computation; everyday math knowledge; attitude toward math; identification of problem type schemas; and visual representation.
Results indicated that the students with typical development significantly outperformed the students with ASDs on word problem solving and everyday math knowledge. Correlation analysis showed that word problem solving performance of the students with ASDs was significantly associated with sentence comprehension, math vocabulary, computation and everyday math knowledge, but that these relationships were strongest and most consistent in the students with ASDs. No significant associations were found between word problem solving and attitude toward math, identification of schema knowledge, or visual representation for either diagnostic group. Additional analyses suggested that everyday math knowledge may account for the differences in word problem solving performance between the two diagnostic groups.
Furthermore, the students with ASDs had qualitatively and quantitatively weaker structure of everyday math knowledge compared to the typical students.
The theoretical models of the linguistic approach and the schema approach offered some possible explanations for the word problem solving difficulties of the students with ASDs in light of the current findings. That is, if a student does not have an adequate level of everyday math knowledge about the situation described in the word problem, he or she may have difficulties in constructing a situation model as a basis for problem comprehension and solutions. It was suggested that the observed difficulties in math word problem solving may have been strongly associated with the quantity and quality of everyday math knowledge as well as difficulties with integrating specific math-related everyday knowledge with the global text of word problems.
Implications for this study include a need to develop mathematics instructional
approaches that can teach students to integrate and extend their everyday knowledge from real-life contexts into their math problem-solving process. Further research is needed to confirm the relationships found in this study, and to examine other areas that may affect the word problem solving processes of students with ASDs.
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TABLE OF CONTENTS
CHAPTER I. INTRODUCTION
Background and Need...1
Word Problem Solving.. ... 5
Definitions... ...7
Statement of the Problem ... 11
CHAPTER II. REVIEW OF LITERATURE ... 13
Schematic Approaches to Word Problem Solving ... 14
Schema Induction Theory and Analogical Thinking ... 15
Semantic Relations... 16
Marshall's Schema Theory ... 17
Summary ... 19
Linguistic Approaches to Word Problem Solving ...20
Classic Models of Comprehension Process ... 21
Situation Model ... 24
Mental Model ... 25
Brain Imaging Studies and Word Problem Solving ... 26
Summary ... 29
Research Studies on Factors Associated with Word Problem Solving ...29
Word Reading/Decoding ... 29
Sentence Comprehension ... 31
Mathematics Vocabulary ... 34
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Everyday Math Knowledge ... 39
Attitude toward Math ... 40
Problem Type Schema Knowledge ... 43
Visual Representation ... 44
Summary ... 48
Characteristics of Students with ASDs ... 49
Cognitive Theories Explaining High Functioning ASD ... 49
IQ and Overall Academic Abilities of Students with ASDs. ...53
Sentence Comprehension, Word Reading, Semantics and Everyday Knowledge ...56
Visuospatial Abilities ...58
Mathematics Abilities of Students with ASDs...60
Summary and Rational ... 62
Research Questions ... 66
CHAPTER III. METHOD ... 68
Participants ... 68 Inclusion Criteria ... 69 Recruiting Procedures ... 70 Research Design ... 72 Measures ... 72 Screening... 72 Dependent Variables ... 75 Procedures ... 82 Background Information ... 82
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Administration of Instruments ... 82
Administration Sequence ... 82
Data Analysis ... 83
CHAPTER IV. RESULTS ... 84
Preliminary Analysis ... 84 Descriptive Statistics ... 84 Main Analysis ... 85 Research Question 1 ... 85 Research Question 2 ... 88 Research Question 3 ... 89 Research Question 4 ... 91 Additional Analysis ... 92
Analyses on Role of the Significant Factor in Differences ... ...93
Additional Analysis for Everyday Math Knowledge ... 93
Analysis for Visual Representation... ...95
Summary of Results ... 99
CHAPTER V. DISCUSSION ... ...101
Group Differences in Word Problem Solving and Associated Factors...102
Factors Associated with Math Word Problem Solving...104
Everyday Math Knowledge... 105
Other Factors Associated with Word Problem Solving.. ... 108
Visual Representation ... 110
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Limitations ... 113
Future Research ... 114
Conclusion ... 115
REFERENCES ... 118
APPENDIX A [Informed Consent] ... 135
APPENDIX B [MWPS by Problem Schema Type] ... 140
APPENDIX C [Problem Type Schema Finder- Student Form (PTSF-ST)]... 141
APPENDIX D [Visual Representation Observation Form]... 142
APPENDIX E [Correlation Matrix for Word Problem Solving Variables: Students with ASDs] ... 143
APPENDIX F [Correlation Matrix for Word Problem Solving Variables: Typical Students] ... 144
APPENDIX G [Visual Representation: Student Sample Category 1] ... 145
APPENDIX H [Visual Representation: Student Sample Category 2] ... 146
APPENDIX I [Visual Representation: Student Sample Category 3] ... 147
APPENDIX J [Visual Representation: Student Sample Category 4] ... 148
APPENDIX K [Visual Representation: Student Sample Category 5] ... 149
APPENDIX L [Visual Representation: Student Sample Category 6] ... 150
APPENDIX M [Visual Representation: Student Sample Category 7] ... 151
APPENDIX N [School Districts' Permission to Recruit Participants] ... 152
APPENDIX O [ Permission to Use Facility] ... 154
APPENDIX P [ Recruiting/Advertising through Asperger Syndrome and High Functioning Autism Society of New York (AHANY)] ... 155
APPENDIX Q [ Everyday Math Knowledge and Concepts Included in the Test Items in TOMA2-GI] ... 156
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APPENDIX R [Every Day Math Knowledge Included in the Test Items in TOMA2-SP] ... 157
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LIST OF TABLES
Table Page
1. Demographic Profile of Participants 68
2. Instruments and Corresponding Variables 73
3. Comparison of Students with ASDs and Students with Typical Development on Age and IQ
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4. Comparisons of Means and Standard Deviations for Factors Related to Word Problem Solving According to the Two Diagnostic Groups
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5. Crosstabulation of Diagnostic Group and Use of Visual Representation 87 6. Summary of Spearman Correlations ( ) between Word Problem-Solving
Performance and Related Factors for Students with ASDs and Students with Typical Development
89
7. Percent of Problem Representation Used in Correct and Incorrect
Responses of Students With ASDs and Students with Typical Development
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LIST OF FIGURES
Figure Page
Figure 1. Hybrid Model of Schema in Problem Solving Process 18 Figure 2. Number of Responses on Visual Representation Observation Form for
Students With ASDs and Typical Students
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ACKNOLOWDEGEMENTS
This dissertation would not have been possible without the guidance and the help of many people who in one way or another contributed and extended their valuable assistance in the preparation and completion of this study.
First and foremost, I would like to express the deepest gratitude to my advisor, Professor Linda Hickson, who has been the exemplar of scholarly precision, persistence and support. Throughout this process, Professor Hickson has taught me the invaluable lesson - what it means to be a researcher. Without her countless readings of my draft, insightful suggestions and guidance, it would not have been possible for me to come this far.
I would like to express my sincere gratitude to Professor Hsu-Min Chiang, for giving me her time, encouraging and inspiring me to pursue this study, and providing a new perspective on my research idea.
I also would like to thank the committee members, Professor Robert Kretschmer,
Professor Young-Sun Lee, and Professor Victoria Rizzo, for their willingness to give their time and guidance throughout this process.
I sincerely thank to Allison Strand, Executive Director of Special Education, and the teachers in the Half Hollow Hills School District, and Karen Ellis, Executive Director of Special Education at the Nassau BOCES for the support and help that they have provided me to find participants and schedule the numerous testing sessions in the district school buildings.
To Pat Schissel, Executive Director of the Asperger Syndrome and High Functioning Autism Association of New York, Julia Park, Esq., Jennifer Lim, LMSW, and the parents and students who have helped me to complete this study. I cannot find words to express my gratitude to these parents and students. The opportunity to spend one on one time with each participant
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and the parents allowed for me to gain a depth of understanding that I would not have otherwise experienced.
Finally, to my family, I owe my deepest gratitude to my husband Kunil, and my children Eugene and Devin who have provided enduring support throughout the years. Their
encouragement, smiles and hugs have given me the strength to continue and keep everything in perspective.
1 CHAPTER I
INTRODUCTION Background and Need
The ability to solve mathematical word problems has long been recognized as an essential component of math competency. The National Council of Teachers of Mathematics (NCTM, 2000) stated that problem solving should be the focus of mathematics teaching because it encompasses skills and functions which are an important part of everyday life. The NTCM also states (Principles and Standards for School Mathematics, 2000), "Good problems give students the chance to solidify and extend their knowledge and to stimulate new learning. Most
mathematical concepts can be introduced through problems based on familiar experiences coming from students' lives or from mathematical contexts." However, problem solving is a challenging task for many young students, especially for students with cognitive difficulties because it requires not only mathematics skills, but also reading comprehension, reasoning, and the ability to transform words and numbers into the appropriate operations (Neef, Nelles, Iwata, & Page, 2003). During the last decade, research efforts to improve teaching and learning in mathematical word problem solving for students with disabilities have been focused on students with learning disabilities (LDs). As a result, various instructional approaches have been
introduced to improve the word problem solving performance of students with LDs (e.g., Fuchs & Fuchs, 2002; Griffin & Jitendra, 2009; Jitendra, Griffin, Deatline-Buchman, & Sczesniak, 2007a; Jitendra et al., 2007b; Montague & van Garderen, 2003). Yet, mathematical word
problem solving in students with autism spectrum disorders (ASDs) has rarely been investigated in-depth despite the serious increase in the prevalence of this student population.
2 ASDs are a part of the broader category of pervasive developmental disorders (PDDs), that include Autistic Disorder, Asperger Syndrome (AS), and Pervasive Developmental
Disorder-Not Otherwise Specified (PDD-NOS)[Centers for Disease Control(CDC), 2012]. The defining characteristics of ASDs are qualitative impairments of social interaction and
communication, along with highly focused interests, and restricted and repetitive activities (CDC, 2012). The CDC's most recent data (2012) indicate that an average of one in 88 children has an ASD (based on children who were 8 years old in 2008). The U.S. Department of Education also reported that, from 2007 to 2011, the number of children aged 6 to 21 years receiving services for ASDs under the Individuals with Disabilities Education Act (IDEA) increased from 256,863 to 406,957 in the 50 states of the U.S. (Office of Special Education Programs, Data Analysis System, 2012). The number of students with ASDs who were included in general education classrooms for more than 40% of the school day in 2007 was 135,023 (US Department of Education, 2007). This represented approximately 53% of the total population of students with ASDs who were receiving educational services under the IDEA in 2007 (US Department of Education, 2007).
Because of the current increase in children who receive educational services for autism under IDEA (U.S. Department of Education, 2004), a significant effort has been put forth on the part of federal and state education programs, school districts, educators and families to support children with ASDs. One of the critical mandates of the 1997 and 2004 amendments to IDEA is that students with disabilities must have meaningful access to the general education curriculum. According to the No Child Left Behind Act (NCLB, 2001), students with disabilities are held to the same high academic standards required of all students. This law also requires that schools be accountable for the academic progress of all students, including the achievement of students with
3 disabilities, on statewide assessments of reading and mathematics. As these two powerful laws drive schools to use evidence-based educational interventions for all students, there has been increased attention to academic achievement, especially in literacy and mathematics for students with ASDs (Bouck, 2009). As a result, high-functioning, school-aged children with ASDs are expected to be placed in classrooms with same-aged, typically developing peers, and to be
working toward similar academic goals as these peers (Chiang & Lin, 2007; Estes, Rivera, Bryan, Cali, & Dawson, 2011).
Approximately, 65% of the children, the CDC survey identified as having ASDs, did not have intellectual disabilities (IQ lower than 70). In fact, the largest increases from 2002 to 2008 were for those having IQ scores higher than 70 although there were increases in the identified prevalence of ASDs at all levels of intellectual ability (CDC, 2012). Researchers have suggested that students who have high-functioning ASDs commonly display unique cognitive, social and academic characteristics (Chiang & Lin, 2007; Kenworthy, 2010; Myles and Simpson, 2002). These students exhibit a wide range of academic achievement outcomes, from significantly above average to average or far below average in some areas (Griswold et al. 2002; Mayes & Calhoun, 2008). However, patterns of academic achievement in these high-functioning students with ASDs are not currently well-explored, and the factors associated with positive academic outcomes are not well understood (Estes et al., 2011). In particular, relatively little is known about mathematical word problem solving abilities and the factors associated with variability in mathematical performance in the population with high-functioning ASD.
Chiang and Lin (2007) discussed the IQ profiles and academic achievement of students with high-functioning autism and Asperger Syndrome (i.e., HFA and AS) based on 18 studies that were published between 1986 and 2006. Individuals with autism who have average and above
4 IQ are regarded as having high functioning autism (HFA) (Baron-Cohen, 2000; Chiang & Lin, 2007). Individuals who have normal language development and share the same characteristics as autism in the area of social interaction as well as repetitive and stereotypic patterns of behaviors are said to have Asperger syndrome (AS) (Chiang & Lin, 2007).
Chiang and Lin (2007) found that the majority of the participants in those studies
demonstrated mathematical ability in the average range related to the norm, many with clinically modest mathematical weakness compared to their IQs, and some showing mathematical
giftedness. Chiang and Lin's findings called for more systematic and comprehensive
examinations aimed at providing age-appropriate mathematics curricula for students with ASDs. However, as Chiang and Lin noted, educators tend to focus on the disability of students with ASDs rather than on their actual ability and unique talent. This attitudes may be driven by inconsistent research findings focused on complex relationships between IQ profiles, academic achievement and social functioning (Estes et al., 2011). Therefore, more evidence is needed to ensure that these students receive an appropriate and effective instruction to advance their academic attainment.
Educational equity is one of several principles articulated in the NCTM (2000) and is based on the fundamental notion that all students, “regardless of personal characteristics, backgrounds, or physical challenges” (p. 12) should have access to a curriculum that is challenging (Jitendara & Star, 2011). As the NCTM stresses that problem-solving skills are a critical component of all areas of the mathematics curriculum, the ability to solve mathematical word problems is increasingly essential to academic success (Jitendra et al., 2005). However, currently, there are few published research studies on mathematical word problem solving comparing students with ASDs with students from the typically developing student population.
5 Although Chiang and Lin's study (2007) examined the general mathematical abilities of students with HFA and AS, their study needs to be extended to yield more empirical evidence in the area of school mathematics, such as word problem solving as well as the various factors associated with students' performance in mathematics.
Word Problem Solving
Mathematical problem solving is a complex cognitive activity involving a number of processes and strategies (Mayer, 1999; Montague & van Garderen, 2003) and frameworks (e.g., Hegarty et al., 1995). In this study, the definition of mathematical word problem solving encompasses several components. Word problem solving is: a) a goal directed behavior (Anderson, 2005) to figure out unknown mathematical information in narrative problems; b) a process that requires problem interpretation, representation, plan solution and execution of the plan, not merely computational operations embedded in word form (Mayer, 1985; Montague & Applegate, 1993); c) both single and multiple steps (Fuchs, Fuchs, & Prentice, 2004; Montague & Applegate, 1993); and d) a form of “transfer,” which requires a person to apply the problem solution rules to other narrative problems (Fuchs, 2004; Mawer & Sweller,1985).
Researchers in the field of cognitive psychology have provided helpful paradigms for addressing the complex nature and role of knowledge in students' word problem solving (Carpenter & Morser, 1984; Kintsch & Greeno, 1985; Mayer, 1975). The General Problem Solving (GPS) model which was created in the 1970s (i.e., Newell & Simon, 1972) viewed the human problem solver as an information processing system manipulating symbolic structures. The theoretical framework of this model was the information processing paradigm, which
attempted to explain all behavior as a function of memory operations, control processes and rules (Anderson, 2005). The method for testing the theory involved developing a computer simulation
6 and then comparing the results of the simulation with human behavior on a given task. The research studies within this theoretical framework explored general problem-solving heuristics in domains of elementary logic, chess, and puzzles combined with experimental and theoretical detail (i.e., Anderson, 2005). The GPS also introduced the use of productions as a method for specifying cognitive models. This computer simulation program and the associated theoretical framework made a significant impact on the direction of cognitive psychology and research in mathematical problem solving behaviors (i.e., van Dijk & Kitsch, 1983).
During the 1980s, the focus of the research studies in this area shifted to the crucial role of expertise and domain-specific knowledge and processes in a complete account of problem solving (e.g., Chi, Glaser, & Farr, 1988; Ericsson & Smith, 1991). Most of the theoretical models that emerged during this period viewed word problem solving as a process consisting of two major components: comprehension-representation and problem solution. Theories of problem solving processes have been developed in-depth, as have theories of the processes of language comprehension (Kintsch & Greeno, 1985). These two lines of theory come together in analyses of the representation of problems because text is used to convey problem information or instructions. Subsequent studies (e.g., Campbell, 1992; Hegarty et al, 1995; Mayer, 1989, 1992; Schoenfeld, 1985, 1987) aimed to provide an account of the domain-specific strategies used by successful and unsuccessful problem solvers for solving arithmetic word problems, and how these strategies accounted for individual differences in performance.
The most noteworthy progress in the field was made by approaching word problem solving from two different angles: the schematic and linguistic approaches. In fact, the schematic analysis of arithmetic word problems is interrelated with the linguistic approach (e.g., Carpenter & Moser, 1982; Gick & Holyoak, 1983; Marshall, 1995). Those who adopted the schematic
7 approach were influenced by notions such as 'frames', 'structures', and 'analogies' which emerged from the connectionists' paradigm of information processing research (Rumelhart, 1980;
Thompson, 1985) and the constructivist approach (Reusser, 1992; Vergnaud,1988). Their theoretical endeavor helped categorize word problems in arithmetic and algebra (Nesher, Hershkovitz, & Novotnal, 2003). In the mean time, under the linguistic approach, various constructs were proposed to account for comprehension of word problems. Notable research includes the works of Kintsch and van Dijk (1983), Kintsch and Greeno (1985), and Reusser (1988) who introduced notions such as 'textbase', 'situation model', and 'mental model'.
Numerous research studies concerning general and some disability populations (i.e., LD, intellectual disability) have corroborated the above theoretical paradigms. These models have successfully predicted how students' cognitive characteristics, problem-solving behaviors and instructional factors contribute to their word problem-solving performance (e.g., Hegarty et al., 1992; Jitendra et al., 2009; Judd & Bilsky, 1989; Pape, 2004). In addition, some researchers have attempted to build a paradigm to explain the various factors associated with word problem solving of students with LD (e.g., Fuchs et al., 2006; Nesher et al., 2003). However, it still remains unclear what factors affecting the solution path are actually adopted by a given solver (Nesher et al., 2003). Moreover, due to the paucity of math research with the ASD population, word problem solving performance of students with high-functioning ASDs has not been examined in the context of any models or theoretical frameworks.
Definitions
Autism. The term, autism is used interchangeably with autism spectrum disorders (ASDs) in this study. IDEA (2004) defines autism:
8 Autism means a developmental disability significantly affecting verbal and nonverbal communication and social interaction, generally evident before age three, that adversely affects a child's educational performance. Other characteristics often associated with autism are engagement in repetitive activities and stereotyped movements, resistance to environmental change or change in daily routines, and unusual responses to sensory experiences [34 CFR Section 300.8 (c) (1) (i-iii)].
Autism spectrum disorders (ASDs). According to the CDC (2012), autism spectrum disorders are (ASDs) a group of developmental disabilities that often are diagnosed during early childhood (onset before age 3) and can cause significant social, communication, and behavioral challenges over a lifetime. ASDs include Autistic Disorder, Asperger Syndrome, and Pervasive Developmental Disorder-Not Otherwise Specified (PDD-NOS) (CDC, 2012).
Autistic disorder. Autistic disorder is also called "classic" autism (CDC, 2012).
Individuals with autistic disorder usually have significant language delays with onset before age three. Autistic disorder is marked by three defining features,: (1) impaired social interaction (e.g., lack of social or emotional reciprocity); (2) impaired communication (e.g., delay or total absence of spoken language); and (3) restricted, repetitive, and stereotyped patterns of behavior, interests, and activities (e.g., stereotyped and repetitive motor mannerisms and/or persistent preoccupation with parts of objects). Many individuals with autistic disorder also have intellectual disabilities (CDC, 2012).
Asperger Syndrome (AS). Asperger Syndrome (AS) is also called as "Asperger’s Disorder." Asperger Syndrome is one of the autism spectrum disorders (ASDs) (CDC, 2012).
9 Individuals with Asperger Syndrome usually have some milder symptoms of autistic disorder. However, they do not have a language delay and, by definition, must have an average or above average IQ (measure of intelligence) (CDC, 2012).
Pervasive developmental disorder - Not Otherwise Specified (PDD-NOS). PDD-NOS (also called "atypical autism") refers to individuals who meet some of the criteria for autistic disorder or Asperger Syndrome, but not all, may be diagnosed with PDD-NOS. People with PDD-NOS usually have fewer and milder symptoms than those with autistic disorder.
High-functioning autism spectrum disorders (high-functioning ASDs). According to Hochhauser and Engel-Yeger (2010), high-functioning autism spectrum disorders refer to individuals with average or above average intelligence with a diagnosis of Autism, Asperger's syndrome, or pervasive developmental disorders not otherwise specified (PDD-NOS). It has been widely debated how best to approach the definition for autism without intellectual disability (often termed high-functioning autism), Asperger’s Syndrome and PDD-NOS, but a consensus has not emerged (Volkmar & Lord, 2007; Volkmar, State, & Klin, 2009). Although there are no formal diagnostic criteria for high-functioning autism spectrum disorders,
researchers have distinguished those high-functioning cases of autism spectrum disorders by their relative preservation of linguistic (verbal) ability and cognitive development ( Klin & Volkmar, 2003; Volkmar, State, & Klin, 2009; Volkmar & Lord, 2007). Therefore, individuals with high-functioning autism spectrum disorders may function well in literal contexts but they have difficulty using language in a social context. Examples include a lack of comprehension of social situations, lack of initiation and sharing with others mutually and reciprocally ( e.g., Church, 2010; Hochhauser & Engel-Yeger, 2010; Kenworthy, 2010).
10 Semantic structure. Semantics is the part of linguistics which refers to the study of meaning of the words themselves (lexical semantics) and the meaning according to the linguistic context and the speaker (grammatical semantics) (Vogindroukas et al., 2003). In addition, words may convey different meanings according to the way they are used in social contexts and the speaker’s intention (Bishop, 1999). Semantic structure means organization that has meaning. Understanding semantic structure is one of the critical components for word problem solving skills (Canobi, 2009; Carpenter & Moser, 1984; Fuchs, Seethaler, Powell, Hamlett, & Fletcher, 2008; Jitendra & Hoff, 1996).
Semantic relations. In the narrow sense, semantic relations relate to concepts or
meaning. Relations between concepts, senses, or meanings should not be confused with relations between the terms, words, expressions, or signs that are used to express the concepts. A number of research studies reported that semantic relations in word problem structure have more
influence on children's strategies for solving word problems than syntactic components (Morser & Capenter 1982; Griffin & Jitendra, 2008). For instance, Riley et al. (1983) defined that
semantic relations in word problem solving as conceptual knowledge about increases, decreases, combinations, and compare.
Schema. Schema is assigned to various meanings depending on the type of studies and discussions. Marshall (1995) discussed the nature of schema based on Piaget and Bartlett's views, defining schema in general terms as a memory structure that develops from an individual's
experiences, and guides the individual's response to the environment (p.15). Gick and Holyoak (1980) defined a schema in word problem solving as a general description of two or more
problems, which a person uses to group problems into types that require similar solution methods. This study will follow Gick and Holyoak's definition in the discussion of word problem solving.
11 Mathematical word problems. In this study, the term mathematical word problem is interchangeable with arithmetic word problem, story problem, or word problem.
Statement of the Problem
Current federal legislation supports educational equity of all students by highlighting challenging learning standards and school accountability. Within the area of mathematics, K-12 general education curriculum and statewide assessments are increasingly focused on problem solving. However, helping students with disabilities achieve competence in mathematical word problem solving has proven especially challenging because it is related to various aspects of academic and cognitive factors. Relatively little research has been done on the relations among the various factors associated with the mathematical word problem solving of students with ASDs, and whether the cognitive abilities that mediate various aspects of mathematics
performance are shared or distinct. Understanding such relations can provide theoretical insight into the nature of mathematics development and can provide practical guidance about the identification of mathematics difficulties (Fuchs et al., 2006).
For these reasons, the purpose of this study is two-fold. The primary purpose is to examine mathematical word problem solving performance of students with ASDs and their typical peers. Although there have been some research endeavors to examine the mathematics abilities of students with ASDs (i.e., Chiang & Lin, 2007; Mayes & Calhoun, 2008), word problem solving pertinent to the general mathematics curriculum has been underexplored for this population. The research in this field has commonly compared problem solving between typical achievers and high achievers, and to some extent, students with LDs or intellectual disabilities (e.g., Bilsky & Judd, 1986; van Garderen & Montague, 2003). Nonetheless, few studies directly compared word problem solving of students with ASDs to their typically developing peers.
12 Given that the number of students with ASDs is on the rise, such a paucity of research is
problematic. Since mathematical word problem solving is one of the central themes in school mathematics (NCTM, 2000), it is crucial to investigate the word problem solving competence of students with ASDs so that educators can help these students gain equal access to the general education curriculum.
Secondly, this study aims to clarify the factors associated with word problem solving and the solution paths adopted by students with ASDs. Research has been undertaken to determine the characteristics that affect successful math problem solving, including the presence of a disability (Bilsky, Blachman, Chi, Chan, Mui, & Winter, 1986; Fuchs & Fuchs, 2002; Jitendra & Star, 2011; Judd & Bilsky, 1989), knowledge of strategies (Montague, 2008), the type of
problem, (Garcia, Jimenez, & Hess, 2006; Griffin & Jitendra, 2009), irrelevant information (Censabella & Noel, 2008; Passolunghi, Marzocchi, & Fiorillo, 2005), and the ability to visually represent the problem (Booth &Thomas, 2000; Hegarty & Kozhevnikov, 1999; Van Garderen, 2003). The current problem solving models, such as those within the linguistics and schematic approach paradigm, were built based on typical cognitive processing. Thus, it is vital to explore whether or not these models can explain the problem solving processes of students with ASDs in regards to their unique cognitive characteristics.
Math research still lacks clarity on exactly where students with ASDs are confident or struggle in the areas of the general mathematics curriculum. Identifying the word problem-solving abilities of students with ASDs will potentially support educators to implement appropriate instructional programs to meet the students' academic needs.
13 CHAPTER II
REVIEW OF LITERATURE
The purposes of this chapter are, first, to present the literature and research to support the theoretical framework of the study, including schematic and linguistic approaches, and the factors affecting word problem solving performance. A second purpose is to review and make connections between the theories in word problem solving and the relevant cognitive and academic characteristics of students with ASDs.
This chapter begins with a literature review on the two distinctive but closely related paradigms of mathematical problem solving, the schematic and linguistic approaches. As briefly introduced in Chapter 1, the schematic and linguistic approaches were influenced by the early theories in GPS as well as the theories on language comprehension. Although many aspects of these two approaches have been synthesized and embodied in various models of word problem solving, the focus of each paradigm can be distinguished. The first and second sections explain the history of each paradigm, the word problem solving models constructed under these
paradigms, and the important problem solving characteristics defined by the models. Another important aspect of this study pertains to the factors associated with word
problem solving processes. The third section of this chapter reviews the prior research studies on these factors and the extent to which each factor is associated with the word problem-solving performance of students with varying abilities. These factors include word reading, sentence comprehension, mathematics vocabulary, arithmetic computation, everyday math knowledge, attitude toward math, problem type schema knowledge and visual representation. The
14 The fourth section explains the characteristics of students with ASDs by reviewing the theories which have provided the important theoretical frameworks in the field. This section also includes a discussion of research findings on the connections between sentence comprehension, semantics and visual representation in ASDs, and what is known about abilities of students with ASDs in IQ, academics, and mathematics. Finally, the chapter concludes with a summary and rationale, and the list of research questions.
Schematic Approaches to Word Problem Solving
The schema construct offers an account of how old knowledge might influence the acquisition of new knowledge (Anderson, 1977a, 1977b). Schema theory was introduced in the fields of psychology and education by Bartlett (1932), and has received empirical support from studies in psycholinguistics and mathematics education. Bartlett's schema theory described how information might be stored and connected in human memory (Marshall, 1995). Bartlett (1932) proposed that people have schemas or unconscious mental structures that represent an
individual's generic knowledge about the world, such as in things, events and situations. He suggested that people normally reconstruct incoming information based on their own schemas that are comprised of past experiences; thus, incoming information is often added, ignored, or transformed through such an active process, and false memory is considered to be its by-product. If schemas are not formed appropriately, new information remains fragmented; it cannot be integrated into a coherent whole, leading to difficulties in understanding the outer world.
Anderson (e.g., 1977a, 1977b) extensively investigated schema knowledge and schema-directed processes. Anderson (1977b) described a schema as a structure that indicates a typical relationship among its components (p. 3). He also suggested that schemas capture both the patterns of relationships, such as categories, as well as their linkage to operations. That is,
15 schemas conceptually represent categorical knowledge according to a slot structure in which slots specify values of various attributes that members of a category possess (Anderson, 1977a). According to Anderson, therefore, schemas provide a form of representation for complex
knowledge that is important in problem solving processes. Schemas are also an important aspect of expert knowledge (Anderson, 2005). In mathematics, the expert knowledge that underlies the ability to recognize problem categories or types has been characterized as involving the
development of organized conceptual structures, or schemas (Marshall, 1995). Schema Induction Theory and Analogical Thinking
A major challenge in producing mathematical problem-solving expertise is the development of schemas (Fuchs et al., 2004). Gick and Holyoak (1983) introduced schema-induction theory which explains how people induce a general schema from experiences with specific objects or events. Gick and Holyoak suggested that exposure to instances that vary in surface features allow people to form generalized rules that are not restricted to overly
specialized contexts, thus facilitating transfer. In order to induce a general problem solving schema from given examples, it is necessary to know what semantic relations are involved in common and how they differ. Knowledge of word problem-solving patterns can be mapped on the basis of their relational (i.e., problem types and solutions) correspondences. Recognizing such similarity between a target problem and a source problem is a fundamental cognitive process in solving problems and it involves analogical thinking (Mayer, 1996).
The primary nature of analogical thinking is the transfer of knowledge from one situation to another by a process of mapping a set of one-to-one correspondences (often incomplete) between aspects of one body of information and aspects of another (Gick & Holyoak, 1983). Hence, the important assumption of schema theory as it applies to analogical problem solving is
16 that problem schemas are formed through induction as a result of experiencing various instances of the general solution principle or rule (Chen & Mo, 2004). Consequently, the broader the schema (i.e., the more general the description of the problem category), the greater the probability that individuals will recognize connections between novel and familiar problems; thus they will know when to apply the solution methods they have mastered (Fuchs, Seethaler et al., 2008; Gick & Holyoak, 1980; Robins & Mayer, 1993).
Semantic Relations
Carpenter and Moser (1982) identified two basic dimensions which account for the difficulty of mathematical word problems: One is based on syntactic variables, and the other is based on logical structure and the semantic component of the problem. The syntactic dimension includes components such as structural variables concerned with the number of words and positions of the component parts within the problem (Nesher, 1982). The logical structure, which has been incorporated into the semantic component, includes the types of operations involved and the presence or absence of information. The semantic component includes the contextual relationships contributing to problem structure and verbal cue words included in the problem (Nesher, 1982).
A number of research studies reported that semantic relations in word problem structure have more influence on children's strategies for solving word problems than syntactic
components (Carpenter & Morser, 1982; Griffin & Jitendra, 2008). Riley et al. (1983) defined semantic relations in mathematical word problems as "conceptual knowledge about increases, decreases, combinations, and comparisons." The studies under this theoretical framework have demonstrated that instructions focused on semantic relations in word problems produce positive transfer in children's problem-solving strategies (e.g., Carpeter & Morser, 1984; Fuchs et al.,
17 2003). Particularly, early studies of semantic relations were focused on young children's word problem solving of addition and subtraction in the elementary school curriculum (e.g., Carpenter et al., 1982; Heller & Greeno, 1978; Marshall, 1985; Morser et al., 1984; Riley, et al., 1983). These studies examined effects of teaching semantic relations which were commonly conditional to the type of arithmetic operations contained in the problem (Marshall, 1995). For example, Heller and Greeno (1978), Riley et al. (1983) and Marshall (1985) introduced the schema types in word problems in relation to semantic relations in the primary level of addition and
subtraction problems: Change, Combine and Compare. Carpenter and Moser (1982, 1984) also identified six problem schema categories of addition and subtraction word problems: Joining, Separating, Part-part-whole, Comparison, Equalizing-add-on and Equalizing-take away. Marshall's Schema Model
Marshall discussed the nature of schemas in word problem solving extensively (i.e., Marshall, 1995). First, a schema is neither procedural nor declarative knowledge. Procedural knowledge denotes rule knowledge which relates to skill acquisition and performance.
Declarative knowledge is composed of concepts and facts. Both conceptual and rule knowledge are integral parts of a schema; neither alone is sufficient for problem solving (Marshall, 1995). Second, the point at which the schema becomes purposely invoked is when there is an unknown in a situation (e.g., Jose had 36 pennies. He gave some to his friend. Now he has 22 pennies. How many pennies did he give his friend?). In other words, a schema is a goal-oriented cognitive mechanism; the goal is to solve the problem of an unknown. In order to solve a problem, a set of goals or sub-goals needs to be established and procedures need to be identified for achieving them (p.54). Third, the model of word problem solving processes is supported by the four knowledge components of the storage mechanism (See Figure 1).
18
Figure 1. Hybrid model of schema in problem solving process. Cited from Marshall, S. P. (1995). Schemas in problem solving. New York: Cambridge University Press, p.379.
The four knowledge components are identification, elaboration, planning, and execution knowledge. Identification knowledge serves as the recognition of patterns-problem types; the stored details and abstractions assist the individual to confirm that the schema may fit the problem. Elaboration knowledge works in the opposite direction of identification knowledge; it serves to determine whether the problem fits the schema. The individual uses the basic form of the mental model with specific key elements, and the problem supplies the details fitting into these elements. Procedural knowledge serves in formulating plans for solving the problem in sequence in addition to setting goals and selecting operations for obtaining them. Execution knowledge carries out already learned algorithms step by step.
The theoretical foundation of Marshall's model is a hybrid model which adapts two views of human cognitive mechanisms: production systems and neural network models. Examples of
19 production system models include Adaptive Character of Thought (ACT, Anderson,1983) and Symbolic Cognitive Architecture (SOAR, Newell, 1992). An example of a neural networks model (sometimes called connectionist models) is the Parallel Distributed Processing (PDP) model (Rumelhart, McClelland, & the Parallel Distributed Processing Research Group, 1986). Marshall noted that these models could be used to explain her schema model of problem solving as shown in Figure 1; however, the schema itself is not a part of these models. Marshall carried out a series of experiments using this model for word problem solving (Marshall et al., 1987; 1988; 1989), and developed a schema-based instruction model. The model is composed of five type categories (Change, Group, Compare, Restate and Vary) and the four problem-solving knowledge components explained above. Several authors, including Jitendra and colleagues (e.g., 2009, 2011), and Fuchs and colleagues (e.g., 2002, 2008) employed similar schema-based approaches in their intervention studies that resulted in successful instructional applications.
Summary
Since Anderson's seminal work on schema knowledge and schema-directed processes (e.g., 1977a, 1977b), a number of researchers in the area have expanded the schematic approach by adapting various notions such as the theories of schema induction and analogical thinking (e.g., Gick & Holyoak, 1983). For instance, Marshall (1995) extensively investigated schemas and built a theory of schemas in word problem solving. She has identified four types of
knowledge (identification knowledge, elaboration knowledge, planning knowledge, and execution knowledge) which generally corresponds with the models of human cognition mechanism. Most significantly, Marshall's theoretical framework helped many researchers to categorize word problems in arithmetic and algebra (i.e., Change, Group, Compare, Restate and
20 Vary). Indeed, the gist of schematic approaches is problem type categorization - recognition of the patterns or the problem types which are identified by the semantic components encompassing the contextual relationships in the word problem structure (Nesher, 1982). These categorizations were well-established in many published intervention studies, including the work done by Jitendra and colleagues (e.g., 2009, 2011), and Fuchs and colleagues (e.g., 2002, 2008). These researchers have reported positive results from their schema based interventions. Yet, few studies have provided direct and detailed analysis of data regarding the schema knowledge and the problem types.
Linguistic Approaches to Word Problem Solving
The major theoretical frameworks of word problem solving developed in the 1970s and 1980s were driven by the GPS models as discussed above. Although linguistic approaches take into account the notion of schema in word problem solving processes, the emphasis is on analysis of comprehension and representation of word problems (i.e., Cummins et al., 1988; Kintsch & van Dijk, 1983; Kintsch and Greeno 1985; Reusser, 1988), and the strategies used in comprehending word problems. Linguistic theorists who built word problem-solving models using linguistic approaches agreed upon a few assumptions. First, arithmetic word problems could be understood within the framework of the general theory of discourse processing
suggested by van Dijk and Kintsch (1983). Second, comprehension strategies are involved in the construction of multi-layered problem representations. Third, situational understanding has the function of bridging the gap between language comprehension and mathematical problem-solving knowledge (Nesher et al., 2003; Stern & Lehrndorfer, 1992). Lastly, the models generally agree on the assumption that the understanding of word problems in written text
21 requires both bottom-up word recognition processes and top-down comprehension processes
(Verhoeven & Perfetti, 2008).
Classic Models of Comprehension Processes
van Dijk and Kintsch (1983) presented a discourse processing model which significantly influenced word problem solving research emphasizing reading comprehension. Their model is based on the assumption that readers of a text build three different mental representations of the text: first level, a verbatim representation of the text; second level, a semantic representation that describes the meaning of the text; and third level, a situational representation of the situation to which the text refers. The semantic structure of the text, namely "textbase," represents the meaning of the text, and it consists of those elements and relations that are directly derived from the text itself (Kintsch, 1998). According to van Dijk and Kintsch, first, a textbase, is obtained by constructing a coherent conceptual representation of the text, called a microstructure. Then, deriving from the microstructure, a hierarchical macrostructure is established that corresponds to the essential ideas expressed in the text (Kintsch & Greeno, 1985, p. 110). This hierarchical organization allows for a fast and effective search.
van Dijk and Kintsch identified the third level, a situational representation, called a "situation model," as a representation of the content of a text, independent of how the text was formulated. They explained that situation models were necessary to explain issues of reference, coherence, perspective taking, translation, individual differences, memory, reordering effects, problem solving, updating knowledge, and learning. According to their model, a situation model is a component which includes inferences that are made using prior or background or everyday knowledge about the domain of the text information. The prior knowledge referred to in the creation of a situation model is more specific with respect to the content of the text while a
22 general kind of prior knowledge is also needed to create a textbase. In terms of word problem solving, a situation model constructed from the text highlights the important arithmetic relations in the problem. Its structure is adapted to the demands of whatever task the reader expects to perform. In this framework, textbase construction is a strategic process; word problems require the use of special comprehension strategies, which ensure that the text will be organized around mathematical concepts, such as set, rather than around the actor's motivations and goals, as would be appropriate for a narrative.
Using a production-rule model, Kintsch and Greeno (1985) constructed a simulated word problem-solving model for a more thorough analysis of processes of text comprehension than had been provided in earlier investigations of arithmetic word problems. The model was
constructed based on the theory of text comprehension developed by Kintsch and van Dijk (1978) and van Dijk and Kintsch(1983). The model also adapted Riley et al.'s (1983) assumptions about the semantic knowledge required for representing the problems and the processes of operating on the numbers in problems to find the answers (p. 110). They showed that the schematic approach's assumptions of semantic structure and problem-solving processes in arithmetic were compatible with general assumptions about text comprehension in the linguistic approach models.
According to Kintsch and Greeno (1985), the understanding of a word problem leads to the construction of several levels of representation: some of them are textual (text base) and others are situational or high level (situation model and/or problem model) (see Staub & Reusser, 1995; Kintsch, 199; Moreau & Coquin-Viennot, 2003). Kintsch and Greeno proposed that a "problem model" is the single high-level representation in their model, coordinated with the understanding of text (textbase) which specifies the elements that are essential for solving the problem. At this level of representation, only information relevant to problem solving is
23 extracted from the textbase or inferred from knowledge relative to the field. Within this
framework, word problems are understood: a) by the creation of set schemas representing the different states of the problem, and b) by bringing together these sets by subordinate schemas (as cited in Moreau & Coquin-Viennot, 2003, p. 110). A set schema is an abstract structure that is stored in long-term memory, and designed to represent the different states of the problem. A set schema contains four attributes that correspond to the object, the quantity, the specification, and the role slots. The procedures of set creation and bringing them together are carried out by the presence of specific clues in the text: numerical values or specific linguistic expressions (such as ‘how many’ and ‘have’) (Moreau & Coquin-Viennot, 2003). The difficulties for problem-solving are, thus, explained by an error of matching between certain linguistic forms contained in the problem ( i.e., ‘more . . . than’) and the schemas (i.e., comparison schema) (Cummins, 1991; Cummins, Kintsch, Reusser, & Weimer, 1988; Kintsch, 1987; Lewis & Mayer, 1987).
Kintsch and Greeno (1985) also noted another part of representation, the situation model which already had been referred to by Kintsch and van Dijk (1983). Their computer simulation demonstrated that the situation model for a word problem solving task was highly specific, capturing the set relations and arithmetic operations needed for solving the problem. For a more specific explanation of these mechanisms, they adopted the notion of problem schema and hypothesized that it subsumed the situational nature of the problem text (Nathan et al., 1992). In general, the model described the complete reading process, from recognizing words to
constructing a representation of the meaning of the text by including the notion of schema. The emphasis of the model was not only on understanding the meaning of a text but also on a special set of strategies for constructing mental representations of texts that are suitable for applying mathematical operations such as addition and subtraction. In 1988, the model was extended with
24 the so-called construction-integration model (Kintsch, 1988), followed by a completely updated theory ten years later (Kintsch, 1998).
Situation Model
Unlike Kintsch and Greeno's model (1985), which proposed the problem model as the single highest level representation, Reusser (1989), Staub and Reusser (1995), and Nathan, Kintsch and Young (1992) identified the situation model as a representation equally as high as the problem model. Reusser (1995) proposed a model called Situation Problem Solver to provide an analysis of the process of understanding of text and situation. The vital point of this model, compared to the one developed by Kintsch and Greeno (1985), was the construction of a ‘nonmathematical’ representation, the situation model. According to Ruesser, the model proposed by Kintsch and Greeno (1985) relied too much on schema theory which limits its application to simple problems on the mathematical as well as on the verbal and situational levels (Moreau & Coquin-Viennot, 2003).
Nathan et al. (1992) also argued that students may understand a problem in everyday terms but be unable to represent its formal aspects as required for an algebraic solution; therefore, to comprehend a problem, the person must make a correspondence between the formal and his or her own informal understanding of the situation described in the problem.
Nathan et al. suggested that the process of understanding and solving word problems involves three mutually constraining levels of representation that must be constructed by the student: (a) a representation of the textual input itself (the textbase), (b) a model of the situation conveyed by the text in every day terms (the situation model), and (c) the formalization of the situation (the problem model). Akin to the discourse processing model (Kintsch & van Dijk, 1983), this model supposes that the comprehension process begins with forming a propositional textbase when the
25 student reads a word problem, just as with any other text. Then, the textbase is organized into a qualitative situation model, and mapped into a quantitative problem model that captures the algebraic problem structure. Finally, a set of algebraic problem schemas which act as templates for organizing problem-relevant information provides the explicit, graphical cues to guide the construction of these problem models (p. 332). Nathan et al. also noted that the situation model draws on a reader's knowledge of the world to fill in the gaps left by a sparse story (p.333). Within this framework, the difficulties in problem-solving are explained by an error in the understanding of the situation, particularly because it contains many implicit elements and presuppositions (Moreau & Coquin-Viennot, 2003; Nathan et al., 1992). A situation model is, therefore, more qualitative and less formal than a schema (Thevenot et al., 2007). The functional, temporal, and structuring elements described in the text of the problem can be integrated in the situation model, and can influence on individuals’ performance and strategies (Moreau & Coquin-Viennot, 2003; Thevenot et al., 2007).
Mental Model
The internal representations in the thinking process—namely, a mental model—was described by Johnson-Laird (1983) in the domain of text comprehension and reasoning. The mental model can be succinctly defined as a "mechanism whereby humans are able to generate descriptions of system purpose and form, explanations of system functioning and observed system states, and predictions for future states" (Rouse & Morris, 1985, p.7). According to Johnson-Laird and Byrne (1991), individuals use mental models to formulate conclusions, and test the strength of these conclusions by checking whether other models of the premises refute them. This theory is also an alternative view of deductive reasoning that depends on formal rules of inference akin to those of a logical calculus (Johnson-Laird, 1993).
26 Although the mental model was not discussed in the early literature of the linguistic approach, it has started to receive attention lately by the researchers in the area (e.g., Nesher et al., 2003; Stylianou, 2011). Advocates of the mental model framework suggest that solving a problem requires the construction of a mental representation of the situation described by the problem and not by the representations of the problem itself - propositional representations (Nesher et al., 2003). These propositional representations are syntactically-structured strings of symbols, in a mental language, that are derived from reading text. However, rather than
rejecting the notion of propositional representations, the mental model theory treats them as the input to a process that constructs a mental model corresponding to the situation described by the verbal discourse (Johnson-Laird, 1993).
Therefore, the process of deduction, as well as induction (Johnson-Laird, 1993) is carried out on the models rather than on propositional representations. If the solver constructs a mental model of the text to answer a problem, the situation model evoked by the text could have implications for math performance. Consequently, situation models that contradict readers’ expectations about the mathematical operation needed to complete a problem can impair problem solving (Moreau & Coquin-Viennot, 2007). The notions of situation and mental models have provided useful accounts for experts' word problem-solving behaviors in terms of translating and integrating information in word problems and the use of working memory in the process of mental representation (Anderson, 2005; Stylianou, 2011).
Brain Imaging Studies and Word Problem Solving
Recently, brain imaging studies, including results based on functional magnetic resonance imaging (fMRI), have given a new source of information about comprehension and mental representations during mathematical word problem solving processes. Studies in this
27 area have demonstrated that the interaction effects between top-down and bottom-up
perceptual/cognitive functions help to disentangle the function of language-related neural networks (Hanson, Hanson, Halchenko, Matsuka, & Zaimi, 2007). Brain activation differences between text decoding and solving number problems also have been reported by some fMRI studies (e.g., Newman, Willoughby, & Pruce, 2011). Although a detailed discussion about those studies is not within the scope of this study, a few important research developments, such as the brain imaging studies on functional connectivity are briefly discussed in the section.
Brain imaging studies have expanded the investigation of functional connectivity (Friston, 1994) during complex mental activities such as reading comprehension, problem solving or mathematical reasoning. Functional connectivity is a description of the synchronization of activation between remote cortical regions, and it provides a useful characterization of brain activity at the network level (Hanson et al., 2007; Prat, Keller & Just, 2007). In fMRI studies, functional connectivity is measured based on the correlation of the activation time series in pairs of brain areas (Just et al., 2007). This description has been particularly useful for evaluating the response of an intelligent system to task demands, and has provided new insight into the nature of individual differences between such systems (Prat et al., 2007). It has provided evidence that, as task demands increase, functional connectivity also increases as a function of working
memory load (e.g., Diwadkar, Carpenter, & Just, 2000, Prat et al., 2007), reflecting the need for tighter coordination in more demanding conditions.
Prat et al. (2007) investigated the neural bases of individual differences during sentence comprehension by examining the network’s response to two variations in processing demands: a) reading sentences containing words of high versus low lexical frequency (e.g., mistake vs. gaffe), and b) having simpler versus more complex syntax. In an fMRI study, they found that two types
28 of readers, who were independently identified as having high or low working memory capacity in reading tasks, exhibited different levels of synchronization. The results demonstrated greater synchronization in high-capacity readers, in the area between left temporal and left inferior frontal, left parietal, and right occipital regions. This indicated that functional connectivity remained constant or increased with increasing lexical and syntactic demands in high-capacity readers, whereas low-capacity readers either showed no reliable differentiation or a decrease in functional connectivity with increasing demands.
Summary
The models of the linguistic approaches underscore thorough analysis of processes of text comprehension for solving word problems. They were rooted in the general theory of text
comprehension developed by Kintsch and van Dijk (1978) and van Dijk and Kintsch (1983). Various constructs were proposed to account for understanding word problem solving within linguistic approaches, such as text base, problem model, and situation model (e.g., Nathan et al., 1992). The theories in the field have extended and synthesized similar ideas, including the mental model (Johnson-Laird, 1993) and Riley et al.'s (1983) assumptions about the semantic knowledge which is required for representing the problems and the processes of operating on the numbers in problems to find the answers. However, the classic models of the linguistic
approaches do not provide a comprehensive explanation about how other important cognitive mechanisms (e.g., computation skills, knowledge in math, application of strategies, emotional factors and etc.) affect the process of solving word problems. Recently, researchers have been paying attention to mental representation and brain imaging studies to search more clear explanation about how people comprehend text, apply or formulate mathematical concepts and visualize solution for word problems.
29 Research Studies on Factors Associated with Word Problem Solving
The models of the schematic and linguistic approaches have provided a general framework for explaining the processes of mathematical word problem solving. Both approaches commonly focus on text comprehension and understanding of semantic relations presented in word problems. Nevertheless, the models of both paradigms may not clearly explain the relations among the factors associated with word problem solving and whether cognitive abilities mediate the patterns of relationships between word problem-solving performance and related factors. In addition to the two approaches above, this section considers further those factors which have shown various degrees of association with the word problem solving of children with varying abilities (e.g., Anderssen, 2008; Fuchs et al., 2006). The factors discussed in this section are: word reading/decoding; sentence comprehension; math vocabulary;
arithmetic computation; everyday math knowledge; attitude toward math; problem type schemas; and, visual representation.
Word Reading/Decoding
Word reading is an ability measured with letter and word decoding skills through letter identification and word recognition. Research studies have shown that reading or reading-related processes may influence the relations between cognitive abilities and arithmetic (Fuchs,
Compton, & Fuchs, 2005) as well as between cognitive abilities and arithmetic word problems (Swanson & Beebe-Frankenberger, 2004). Most studies related to word reading abilities deal with phonological decoding processing for children with typical development or learning difficulties (e.g., Fuchs et al., 2005). Phonological decoding processing refers to one’s understanding of the sound structure of the language (Fuchs et al., 2005). Many children who have mathematics difficulties also demonstrate reading difficulties (e.g., Fuchs & Fuchs, 2002;
30 Siegel & Ryan, 1989). Thus, it has been suggested that phonological processes, which are
strongly related to reading development, may also be involved in mathematical difficulties (Geary, 1993).
In their 4-year longitudinal study of academic characteristics of mathematics difficulty from first through fourth grade (N=85), Vukovic and Siegel (2010) found that word attack skills and phonological decoding play an important role in mathematics progress of students. However, they noted that their study results did not indicate that students' mathematics difficulty was characterized by deficient phonological skills, but only that these skills were less well developed in the participants with mathematics difficulties than in peer groups.
Murphy, Mazzocco , Hanich and Early (2007) found that on word attack, a measure of non-word reading that taps phonological decoding, the students with math difficulties performed at a lower level than typically developing students from kindergarten to third grade. These findings are consistent with Fuchs et al.’s (2005) study which showed that phonological
processing was a unique predictor of arithmetic fluency in first grade, but not of other aspects of math performance. By contrast, in another study, Fuchs et al. (2006) did not find a direct
relationship between phonological processes and calculation skills at third grade. Taken together, these findings suggest that children with learning difficulties may have lower phonological skills, though they are not necessarily deficient in these skills (Vukovic & Siegel, 2010). However, those extending these results to a discussion of the relation between word reading and arithmetic word problem solving for upper grade students should be cautious since the previous studies assessed phonological processes with a measure of phonological decoding for lower grade students.
31 Sentence Comprehension
In this study, sentence comprehension is defined as an individual's ability to gain
meaning and comprehend ideas and information from written words, and to understand ideas and information contained in written sentences. Mathematics performance and general reading comprehension skills have been shown to be closely related (e.g., Light & DeFries, Hamson, & Hoard, 2000; Jordan, Hanich, & Kaplan, 2003; Jordan, Kaplan, & Hanich, 2002). For example, in a 2-year longitudinal study with 180 elementary students, Jordan et al. (2002) found that
reading difficulties predicted children’s progress in mathematics; whereas, difficulties in mathematics did not predict children’s progress in reading. They also found that when demographic factors (IQ, income, ethnicity, and gender) were held constant, the group with mathematics difficulties progressed at a faster rate in mathematics than the group with reading difficulties. These are consistent results with other studies (e.g., Fuchs & Fuchs, 2002; Jordan & Hanich, 2000) which showed that children with both mathematics and reading deficits performed significantly more poorly on word problem tasks than students with deficits in mathematics only.
Similarly, Vilenius-Tuohimaa, Aunola and Nurmi (2008) examined the association between mathematical word problem-solving performance and reading comprehension skills with 4th grade students (N=225) using path analysis. Children’s text comprehension and
mathematical word problem-solving performance by problem types (compare, change, combine, focus) were tested (See Vilenius-Tuohimaa et al, 2008). Technical reading skills, such as skills in conclusion-interpretation, concept-phrase, cause-effect and main idea were also investigated in order to categorize participants as good or poor readers. The results showed that the covariance between performance on math word problems and reading comprehension was strong